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Notational Conventions

This section provides the notational conventions used throughout this thesis.

A digital image $ {g}=\{{g}_i: i \in \mathbf{R}\}$ on a 2D finite arithmetic lattice $ \mathbf{R}=\{(x,y):0\leqslant x \leqslant
\mathbf{M}-1;0\leqslant y \leqslant \mathbf{N}-1\}$ is a function $ f: \mathbf{R}\rightarrow \mathbf{Q}$ that maps the supporting lattice $ \mathbf{R}$ onto a finite set $ \mathbf{Q}=\{0,1,...Q-1\}$ of signal values (e.g., grey levels, colours, or multi-band signatures). For simplicity, the set $ \mathbf{Q}$ is assumed to be integer-valued in the range between 0 and $ Q-1$.

A pixel in an image $ {g}$ is denoted by $ {g}_i$ where $ i$ is the positioning index in the lattice $ \mathbf{R}$. The set of all pixels in an image $ {g}$ with exception of a pixel $ {g}_i$ is denoted by $ {g}^i$. Similarly, all sites on lattice $ \mathbf{R}$ except the $ i$th site is denoted by $ \mathbf{R}^i$, so that $ {g}^i=\{{g}_j :j
\in \mathbf{R}^i= \mathbf{R}-\{i\} \}$.

The notation $ \mathcal{N}_i$ defines a neighbourhood of a site $ i$, which usually consists of a set of sites with a specific spatial configuration. All sites in a neighbourhood are dependent of each other.

Because the thesis deals mainly with probability texture models, in particular Markov-Gibbs random fields, related terminology and notations are presented below.

A random variable has a nondeterministic value with a given probability distribution. A sequence or array of statistically interrelated random variables form a discrete stochastic process. Let a random variable $ \mathbf{S}_{i}\in\mathbf{Q}$ be associated with each site $ i \in \mathbf{R}$. Then these variables form a random field  $ \mathbf{S}=\{\mathbf{S}_i:
i\in\mathbf{R};\mathbf{S}_{i}\in\mathbf{Q}\}$ in a configuration space  $ \mathcal{S}$ if a probability measure $ Pr(\mathbf{S}:\forall \mathbf{S}\in \mathcal{S})>0$ exists. A random field is a two-dimensional stochastic process. The configuration space has a combinatorial number of different images, $ \vert\mathcal{S}\vert = {Q}^{\vert\mathbf{R}\vert}$.

Under a random field model, an image $ {g}$ is considered as an instance or a sample of the configuration space $ \mathcal{S}$. The image probability $ Pr(S={g})$, or simply, $ Pr({g})$, is known as the joint probability of the image signals. Another commonly-used probability measure is conditional probability of a pixel $ {g}_i$ given all other pixels $ {g}^i$, denoted by $ Pr({g}_i \mid
{g}^i)$. The conditional probability is a local probability of image signals, while the joint probability is a global one.

For convenience's sake, the notational conventions and symbols used in this thesis are summarised in Table 4.1.

Table 4.1: Notational conventions and symbols.
Notation Meaning
$ \mathbf{R}$ A 2D finite arithmetic lattice
$ \mathbf{Q}$ The set of integral signal values
$ {g}$ A digital image defined on $ \mathbf{R}$
$ {g}^i$ All pixels in the image $ {g}$ except the pixel $ {g}^i$
$ \mathbf{S}$ A random field
$ \mathcal{S}$ The configuration space of a random field $ \mathbf{S}$
$ \mathcal{N}_i$ Neighbourhood of the site $ i$
$ \mathbf{R}^i$ The lattice $ \mathbf{R}$ with exception of the $ i$th site
$ \mathbf{S}^i$ All sites of $ \mathbf{S}$ with exception of a site $ S_i$
$ Pr({g})$ Joint probability of a sample image $ {g}$
$ Pr({g}_i \mid
{g}^i)$ Conditional probability of image signal $ g_i$ at pixel $ i$ given $ {g}^i$

next up previous
Next: Texture Analysis and Synthesis: Up: Introduction Previous: Organisation of the Thesis
dzho002 2006-02-22